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# Duality Theory: Biduality in Nonconvex Optimization

### Article Outline

Keywords

Keywords Duality - Nonconvex optimization - d.c. programming - SuperLagrangian - Biduality - Clarke duality - Hamiltonian system

It is known that in convex optimization, the Lagrangian associated with a constrained problem is usually a saddle function, which leads to the classical *saddle Lagrange duality* (i. e. the *monoduality* ) theory. In nonconvex optimization, a so-called superLagrangian was introduced in [1], which leads to a nice biduality theory in convex *Hamiltonian systems* and in the so-called *d.c. programming* .

## SuperLagrangian Duality

*L*(

*x*,

*y*

^{∗}) be an arbitrary given real-valued function on 𝒳 × 𝒴

^{∗}.

*L*: 𝒳 × 𝒴

^{∗}→

**R**is said to be a

*supercritical function*(or a ∂

^{+}-

*function*) on 𝒳 × 𝒴

^{∗}if it is concave in each of its arguments.